The Astronomical Journal, 157:33 (20pp), 2019 January https://doi.org/10.3847/1538-3881/aaef8a © 2019. The American Astronomical Society. All rights reserved.
Deep Exploration of ò Eridani with Keck Ms-band Vortex Coronagraphy and Radial Velocities: Mass and Orbital Parameters of the Giant Exoplanet*
Dimitri Mawet1,2 , Lea Hirsch3,4 , Eve J. Lee5 , Jean-Baptiste Ruffio6 , Michael Bottom2 , Benjamin J. Fulton7 , Olivier Absil8,20 , Charles Beichman2,9,10, Brendan Bowler11 , Marta Bryan1 , Elodie Choquet1,21 , David Ciardi7, Valentin Christiaens8,12,13, Denis Defrère8 , Carlos Alberto Gomez Gonzalez14 , Andrew W. Howard1 , Elsa Huby15, Howard Isaacson4 , Rebecca Jensen-Clem4,22 , Molly Kosiarek16,23 , Geoff Marcy4 , Tiffany Meshkat17 , Erik Petigura1 , Maddalena Reggiani8 , Garreth Ruane1,24 , Eugene Serabyn2, Evan Sinukoff18 , Ji Wang1 , Lauren Weiss19 , and Marie Ygouf17 1 Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA; [email protected] 2 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA 3 Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA 4 University of California, Berkeley, 510 Campbell Hall, Astronomy Department, Berkeley, CA 94720, USA 5 TAPIR, Walter Burke Institute for Theoretical Physics, Mailcode 350-17, Caltech, Pasadena, CA 91125, USA 6 Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA 7 NASA Exoplanet Science Institute, Caltech/IPAC-NExScI, 1200 East California Boulevard, Pasadena, CA 91125, USA 8 Space sciences, Technologies & Astrophysics Research (STAR) Institute, Université de Liège, Allée du Six Août 19c, B-4000 Sart Tilman, Belgium 9 Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA 10 NASA Exoplanet Science Institute, 770 S. Wilson Avenue, Pasadena, CA 911225, USA 11 McDonald Observatory and the University of Texas at Austin, Department of Astronomy, 2515 Speedway, Stop C1400, Austin, TX 78712, USA 12 Departamento de Astronomía, Universidad de Chile, Casilla 36-D, Santiago, Chile 13 Millennium Nucleus “Protoplanetary Disks in ALMA Early Science,” Chile 14 Université Grenoble Alpes, IPAG, F-38000 Grenoble, France 15 LESIA, Observatoire de Paris, Meudon, France 16 University of California, Santa Cruz; Department of Astronomy and Astrophysics, Santa Cruz, CA 95064, USA 17 IPAC, California Institute of Technology, M/C 100-22, 770 S. Wilson Ave, Pasadena, CA 91125, USA 18 Institute for Astronomy, University of Hawai’i at Manoa, Honolulu, HI 96822, USA 19 Institut de Recherche sur les Exoplanètes, Dèpartement de Physique, Universitè de Montrèal, C.P. 6128, Succ. Centre-ville, Montréal, QC H3C 3J7, Canada Received 2018 May 1; revised 2018 October 16; accepted 2018 October 29; published 2019 January 3
Abstract We present the most sensitive direct imaging and radial velocity (RV) exploration of ò Eridani to date. ò Eridani is an adolescent planetary system, reminiscent of the early solar system. It is surrounded by a prominent and complex debris disk that is likely stirred by one or several gas giant exoplanets. The discovery of the RV signature of a giant exoplanet was announced 15 yr ago, but has met with scrutiny due to possible confusion with stellar noise. We confirm the planet with a new compilation and analysis of precise RV data spanning 30 yr, and combine it with upper limits from our direct imaging search, the most sensitive ever performed. The deep images were taken in the Ms band (4.7 μm) with the vortex coronagraph recently installed in W.M. Keck Observatory’s infrared camera NIRC2, which opens a sensitive window for planet searches around nearby adolescent systems. The RV data and direct imaging upper limit maps were combined in an innovative joint Bayesian analysis, providing new ò +0.38 constraints on the mass and orbital parameters of the elusive planet. Eridani b has a mass of 0.78-0.12 MJup and is orbiting ò Eridani at about 3.48±0.02 au with a period of 7.37±0.07 yr. The eccentricity of ò Eridani b’s orbit is +0.06 0.07-0.05, an order of magnitude smaller than early estimates and consistent with a circular orbit. We discuss our findings from the standpoint of planet–disk interactions and prospects for future detection and characterization with the James Webb Space Telescope. Key words: planet–disk interactions – planets and satellites: dynamical evolution and stability – planets and satellites: gaseous planets – stars: planetary systems – techniques: high angular resolution – techniques: radial velocities Supporting material: machine-readable tables
1. Introduction ò Eridani is an adolescent (200–800 Myr)(Fuhrmann 2004; ) ( ) * Mamajek & Hillenbrand 2008 K2V dwarf star Table 1 .Ata Based on observations obtained at the W. M. Keck Observatory, which is distance of 3.2 pc, ò Eridani is the tenth closest star to the Sun, operated jointly by the University of California and the California Institute of Technology. Keck time was granted for this project by Caltech, the University which makes it a particularly attractive target for deep planet of Hawai’i, the University of California, and NASA. searches. Its age, spectral type, distance, and consequent 20 F.R.S.-FNRS Research Associate. apparent brightness (V=3.73 mag) make it a benchmark 21 Hubble Postdoctoral Fellow. 22 system, as well as an excellent analog for the early phases of Miller Fellow. 23 ’ ò NSF Graduate Research Fellow. the solar system s evolution. Eridani hosts a prominent, 24 NSF Astronomy and Astrophysics Postdoctoral Fellow. complex debris disk, and a putative Jupiter-like planet.
1 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al.
Table 1 Booth et al. (2017) used the Atacama Large Millimeter/ Properties of ò Eridani submillimeter Array (ALMA) to image the northern arc of ( < ″) Property Value References the outer ring at high angular resolution beam size 2 . The 1.34 mm continuum image has a low signal-to-noise ratio ( ) ( ) ( ) R.A. hms 03 32 55.8 J2000 van Leeuwen 2007 (S/N) but is well-resolved, with the outer ring extending from Decl. (dms) −09 27 29.7 (J2000) van Leeuwen (2007) ( ) 62.6 to 75.9 au. The fractional outer disk width is comparable Spect. type K2V Keenan & McNeil 1989 ’ ( ) ± ( ) to that of the solar system s Kuiper Belt and makes it one of the Mass Me 0.781 0.005 Boyajian et al. 2012 ; Distance (pc) 3.216±0.0015 van Leeuwen (2007) narrowest debris disks known, with a width of just 12 au. = °± ° V mag 3.73 Ducati (2002) The outer ring inclination is measured to be i 34 2 , K mag 1.67 Ducati (2002) consistent with all previous estimates using lower-resolution L mag 1.60 Cox (2000) submm facilities (Greaves et al. 1998, 2005; Backman et al. M mag 1.69 Cox (2000) 2009). No significant emission is detected between ∼20 and Age (Myr) 200–800 Mamajek & Hillenbrand (2008) ∼60 au (see Booth et al. (2017), their Figure 5), suggesting a large clearing between the inner belt(s) within 20 au and the outer belt outward of 60 au. Booth et al. (2017) find tentative evidence for clumps in the ring, and claim that the inner and 1.1. ò Eridani’s Debris Disk outer edges are defined by resonances with a planet at a The disk was first detected by the Infrared Astronomical semimajor axis of 48 au. The authors also confirm the previous Satellite (Aumann 1985) and later on by the Infrared Space detection of unresolved mm emission at the location of the star Observatory (Walker & Heinrichsen 2000). It was first imaged that is above the level of the photosphere and attribute this by the Submillimetre Common-User Bolometer Array excess to stellar chromospheric emission, as suggested by (SCUBA) at the James Clerk Maxwell Telescope by Greaves MacGregor et al. (2015). However, the chromospheric et al. (1998). ò Eridani is one of the “fabulous four” Vega-like emission cannot reproduce the infrared excess seen by Spitzer debris disks and shows more than 1 Jy of far-infrared (FIR) and SOFIA. excess over the stellar photosphere at 60–200 μm and a lower Finally, recent 11 μm observations with the Large Binocular significance excess at 25 μm (Aumann 1985). Using the Spitzer Telescope Interferometer (LBTI) suggest that warm dust is Space Telescope and the Caltech Submillimeter Observatory present within ∼500 mas (or 1.6 au) from ò Eridani (Ertel et al. (CSO) to trace ò Eridani’s spectral energy distribution (SED) 2018). The trend of the detected signal with respect to the from 3.5 to 350 μm, the model presented in Backman et al. stellocentric distance also indicates that the bulk of the (2009) paints a complex picture. ò Eridani’s debris disk is emission comes from the outer part of the LBTI field of view, composed of a main ring at 35–90 au, and a set of two narrow and hence is likely associated with the dust belt(s) responsible inner dust belts inside the cavity delineated by the outer ring: for the 15–38 μm emission detected by Spitzer and SOFIA. one belt with a color temperature T;55 K at approximately 20 au, and another belt with a color temperature T;120 K at 1.2. ò Eridani’s Putative Planet approximately 3 au (Backman et al. 2009). The authors argue that, to maintain the three-belt system around ò Eridani, three Hatzes et al. (2000) demonstrated that the most likely shepherding planets are necessary. explanation for the observed decade-long radial velocity (RV) Using Herschel at 70, 160, 250, 350, and 500 μm, Greaves variations was the presence of a ;1.5 MJ giant planet with a et al. (2014) refined the position and the width of the outer belt period P=6.9 yr (;3 au orbit) and a high eccentricity to be 54–68 au, but only resolved one of the inner belts at (e=0.6). While most of the exoplanet community seems to 12–16 au. More recently, Chavez-Dagostino et al. (2016) used have acknowledged the existence of ò Eridani b, there is still a the Large Millimetre Telescope Alfonso Serrano (LMT) at possibility that the measured RV variations are due to stellar 1.1 mm and resolved the outer belt at a separation of 64 au. activity cycles (Anglada-Escudé & Butler 2012; Zechmeister Emission is detected at the location of the star in excess of the et al. 2013). Backman et al. (2009) rightly noted that a giant photosphere. The angular resolution of the 1.1 mm map, planet with this orbit would quickly clear the inner region not however, is not sufficient to resolve the inner two warm belts of only of dust particles but also the parent planetesimal belt Backman et al. (2009). MacGregor et al. (2015) used the needed to resupply them, inconsistent with their observations. Submillimeter Array (SMA) at 1.3 mm and the Australia Telescope Compact Array (ATCA) at 7 mm to resolve the outer 1.3. This Paper ring and measure its width (measurement now superseded by ALMA, see below). The data at both mm wavelengths show In this paper, we present the deepest direct imaging excess emission, which the authors attribute to ionized plasma reconnaissance of ò Eridani to date, a compilation of precision from a stellar corona or chromosphere. RV measurements spanning 30 years, and an innovative joint Su et al. (2017) recently presented 35 μm images of ò Eridani Bayesian analysis combining both planet detection methods. obtained with the Stratospheric Observatory for Infrared Our results place the tightest constraints yet on the planetary Astronomy (SOFIA). The inner disk system is marginally mass and orbital parameters for this intriguing planetary resolved within 25 au. Combining the 15–38 μm excess system. The paper is organized as follows: Section 2 describes spectrum with Spitzer data, Su et al. (2017) find that the the RV observations and data analysis, Section 3 describes the presence of in situ dust-producing planetesimal belt(s) is the high-contrast imaging observations and post-processing, most likely source of the infrared excess emission in the inner Section 4 presents our nondetection and robust detection limits 25 au region. However, the SOFIA data are not constraining from direct imaging, additional tests on the RV data, and our enough to distinguish one broad inner disk from two narrow joint analysis of both data sets. In Section 5, we discuss our belts. findings, the consequences of the new planet parameters on
2 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al. planet–disk interactions, and prospects for detection with the additional RV data on ò Eridani. RV data were also collected James Webb Space Telescope, before concluding in Section 6. using the HARPS spectrograph, also on the 3.6 m telescope at La Silla Observatory, during 2004–2008. Together, these data sets were published in Zechmeister et al. (2013). 2. Doppler Spectroscopy In this section, we present our new compilation and analysis 2.2. RV Data Analysis of Doppler velocimetry data of ò Eridani spanning 30 yr. All new spectroscopic observations from Keck/HIRES and APF/Levy were reduced using the standard CPS pipeline 2.1. RV Observations (Howard et al. 2010). The iodine-free template spectrum was ò Eridani has been included in planet search programs at both deconvolved with the instrumental PSF and used to forward Keck Observatory using the HIRES Spectrometer (Howard model each observation’s relative RV. The iodine lines et al. 2010) and at Lick Observatory using the Automated imprinted on the stellar spectrum by the iodine cell were used Planet Finder (APF) and Levy Spectrometer. ò Eridani was as a stable wavelength calibration, and the instrumental PSF observed on 206 separate nights with HIRES and the APF, over was modeled as the sum of several Gaussians (Butler et al. the past 7 yr. 1996). Per the standard CPS RV pipeline, each spectrum was Keck/HIRES (Vogt et al. 1994) RV observations were divided into approximately 700 spectral “chunks” for which the obtained starting in 2010, using the standard iodine cell RV was individually calculated. The final RV and internal configuration of the California Planet Survey (CPS)(Howard precision were calculated as the weighted average of each of et al. 2010). During the subsequent 7 yr, 91 observations were these chunks. Chunks with RVs that are more consistent across taken through the B5 or C2 deckers (0 87×3 5, and observations are given higher weight. Those chunks with a 0 87×14″, respectively), yielding a spectral resolution of larger scatter are given a lower weight. The RV observations R≈55,000 for each observation. Each measurement was taken from Keck/HIRES and APF/Levy are listed in Tables 5 and 6, through a cell of gaseous molecular iodine heated to 50°C, respectively. These data are not offset-subtracted to account for which imprints a dense forest of iodine absorption lines onto different zero-points, and the uncertainties reported in the the stellar spectrum in the spectral region of 5000–6200 Å. This tables reflect the weighted standard deviations of the chunk-by- iodine spectrum was used for wavelength calibration and as a chunk RVs and do not include systematic uncertainty such as PSF reference. Each RV exposure was timed to yield a per- jitter. pixel S/Nof200 at 550 nm, with typical exposure times of In combination among HIRES, the APF, and the other only a few seconds due to the brightness of the target. An instruments incorporated, a total of 458 high-precision RV iodine-free template spectrum was obtained using the B3 observations have been taken, over an unprecedented time decker (0 57×14″, R≈72,000) on 2010 August 30. baseline of 30 yr. We note that the literature RV data included RV observations using the APF and Levy Spectrograph in this study were analyzed using a separate RV pipeline, which (Radovan et al. 2014; Vogt et al. 2014) were taken starting in in some cases did not include a correction for the secular late 2013. The APF is a 2.4 m telescope dedicated to acceleration of the star. Secular acceleration is caused by the performing RV detection and follow-up of planets and planet space motion of a star and depends on its proper motion and candidates, also using the iodine cell method of wavelength distance. For ò Eridani, we calculate a secular acceleration calibration. APF data on ò Eridani were primarily taken through signal of 0.07 ms−1 yr−1 (Zechmeister et al. 2009). This is the W decker (1″×3″), with a spectral resolution of R≈ accounted for in the HIRES and APF data extraction, and most 110,000. Exposures were typically between 10 and 50 s long, likely in the Lick pipeline, but is not included in the reduction yielding S/N per-pixel of 140. The typical observing strategy for the CES and HARPS data. However, the amplitude of this at the APF was to take three consecutive exposures and then effect is much smaller than the amplitude of RV variation bin them to average over short-term fluctuations from stellar observed in the RVs due to stellar activity and the planetary oscillations. On some nights, more than one triple-exposure orbit. Because each of the CES and HARPS data sets cover less was taken. These were binned on a nightly timescale. An than 10 yr, we expect to see less than 1 m s-1 variation across iodine-free template consisting of five consecutive exposures each full data set. We tested running the analysis with and was obtained on 2014 February 17 using the N decker without applying these corrections. All of the resulting orbital (0 5×8″) with resolution R≈150,000. Both the APF and parameters were consistent to within ∼0.1σ. We therefore HIRES RVs were calibrated to the solar system barycenter and neglect this correction in our RV analyses. From the combined corrected for the changing perspective caused by the high RV data set, a clear periodicity of approximately 7 yr is proper motion of ò Eridani. evident, both by eye and in a periodogram of the RV data. We In addition to the new HIRES and APF RV data, we assess this periodicity in Section 4.2. incorporate previously published data from several telescopes ò Eridani’s youth results in significant stellar magnetic into this study. High-precision RV observations of ò Eridani activity. Convection-induced motions on the stellar surface were taken with the Hamilton spectrograph at Lick Observatory cause slight variations in the spectral line profiles, leading to starting in 1987, as part of the Lick Planet Search program. variations in the inferred RV that do not reflect motion caused They were published in the catalog of Fischer et al. (2014), by a planetary companion. As a result, stellar magnetic activity along with details of the instrumental setup and reduction may mimic the RV signal of an orbiting planet, resulting in procedure. false positives. We therefore extract SHK values from each of The Coudé Echelle Spectrograph at La Silla Observatory the HIRES and APF spectra taken for ò Eridani. SHK is a was used, first with the Long Camera (LC) on the 1.4 m measure of the excess emission at the cores of the Ca II H and telescope from 1992 to 1998, then with the Very Long Camera K lines due to chromospheric activity; it correlates with stellar (VLC) on the 3.6 m telescope from 1998 to 2006, to collect magnetic activity, such as spots and faculae, which might have
3 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al.
Table 2 Observing log for NIRC2 Imaging Data
Properties Value Value Value UT date (yyyy mm dd) 2017-01-09 2017-01-10 2017-01-11 UT start time (hh: 05:11:55 05:12:47 05:48:08 mm:ss) UT end time (hh: 09:14:11 09:31:09 08:36:14 mm:ss) Discr. Int. Time (s) 0.5 LL Coadds 60 LL Number of frames 210 260 154 Total integration 6300 7800 4620 time (s) Plate scale (mas/pix) 9.942 (“narrow”) LL Total FoV r;5″ (vortex LL mount) Filter Ms [4.549, LL 4.790] μm Coronagraph Vortex (AGPM) LL Lyot stop Inscribed circle LL 0.5 μm DIMM see- 0.52 0.64 0.97 ing (″) Figure 1. Final reduced image of ò Eridani, using PCA, and 120 principal Par. angle start-end (°) −36–+52 −35–+55 −20–+46 components in the PSF reconstruction. The scale is linear in analog to digital units (ADU).
( ) effects on the RVs extracted from the spectra (Isaacson & SCExAO Jovanovic et al. 2015 . The alignment of the star onto Fischer 2010). the coronagraph center, a key to high contrast at small angles, was performed using the quadrant analysis of coronagraphic images for tip-tilt sensing (QACITS)(Huby et al. 2015, 2017). 3. High-contrast Imaging The QACITS pointing control uses NIRC2 focal-plane corona- Here, we present our new deep, direct, high-contrast imaging graphic science images in a closed feedback loop with the observations and data analysis of ò Eridani using the Keck Keck adaptive optics tip-tilt mirror (Huby et al. 2017;Mawet NIRC2 vortex coronagraph. et al. 2017; Serabyn et al. 2017). The typical low-frequency centering accuracy provided by QACITS is ;0.025λ/D rms, or ;2 mas rms. 3.1. High-contrast Imaging Observations All of our observations were performed in vertical angle We observed ò Eridani over three consecutive nights in 2017 mode, which forces the AO derotator to track the telescope pupil January (see Table 2). We used the vector vortex (following the elevation angle) instead of the sky, effectively coronagraph installed in NIRC2 (Serabyn et al. 2017), the allowing the field to rotate with the parallactic angle, enabling near-infrared camera and spectrograph behind the adaptive angular differential imaging (ADI)(Marois et al. 2006). optics system of the 10 m Keck II telescope at W.M. Keck Observatory. The vortex coronagraph is a phase-mask 3.2. Image Post-processing coronagraph enabling high-contrast imaging at very small angles close to the diffraction limit of the 10 m Keck telescope After correcting for bad pixels, flat-fielding, subtracting sky at 4.67 μm (;0 1). The starlight suppression capability of the background frames using principal component analysis (PCA), vortex coronagraph is induced by a 4π radian phase ramp and co-registering the images, we applied PCA (Soummer et al. wrapping around the optical axis. When the coherent 2012; Gomez Gonzalez et al. 2017) to estimate and subtract the adaptively corrected point spread function (PSF) is centered post-coronagraphic residual stellar contribution from the on the vortex phase singularity, the on-axis starlight is images. We used the open-source Vortex Image Processing— redirected outside the geometric image of the telescope pupil VIP25—software package (Gomez Gonzalez et al. 2017), and formed downstream from the coronagraph, where it is blocked applied PCA on the combined data from all three nights, by means of an undersized diaphragm (the Lyot stop). The totaling more than 5 hr of open shutter integration time (see vector vortex coronagraph installed in NIRC2 was made from a Table 2 for details). We used a numerical mask 2λ/D in radius circularly concentric subwavelength grating etched onto a to occult the bright stellar residuals close to the vortex synthetic diamond substrate (Annular Groove Phase Mask coronagraph inner working angle. coronagraph or AGPM)(Mawet et al. 2005; Vargas Catalán The final image (Figure 1) was obtained by pooling all three et al. 2016). nights together in a single data set totaling 624 frames. The PSF Median 0.5 μm DIMM seeing conditions ranged from 0 52 was reconstructed by using 120 principal components and to 0 97 (see Table 2). The adaptive optics system provided projections on the 351×351 pixel frames excluding the excellent correction in the Ms-band ([4.549, 4.790] μm) with a central numerical mask (2λ/D in radius). This number of Strehl ratio of about 90% (NIRC2 quicklook estimate), similar principal components was optimized to yield the best final to the image quality provided at shorter wavelengths by extreme contrast limits in the 1–5 au region of interest, optimally trading adaptive optics systems such as the Gemini Planet Imager (GPI, Macintosh et al. 2014), SPHERE (Beuzit et al. 2008),and 25 https://github.com/vortex-exoplanet/VIP
4 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al. off speckle noise and self-subtraction effects. The final image (Figure 1) does not show any particular feature and is consistent with whitened speckle noise.
4. Analysis In this section, we present our nondetection and robust detection limits from direct imaging, additional tests on the RV data, as well as our joint analysis of both data sets.
4.1. Robust Detection Limits from Direct Imaging Following Mawet et al. (2014), we assume that ADI and PCA post-processing whiten the residual noise in the final reduced image through two complementary mechanisms. First, PCA removes the correlated component of the noise by subtracting off the stellar contribution, revealing underlying independent noise processes such as background, photon Poisson noise, readout noise, and dark current. Second, the ADI frame combination provides additional whitening due to the field rotation during the observing sequence and subsequent derotation, and by virtue of the central limit theorem, regardless Figure 2. S/N map for our most sensitive reduction, using 120 principal of the underlying distribution of the noise (Marois et al. components. We used the S/N map function implemented in open source 2006, 2008). Henceforth, we assume Gaussian statistics to package VIP (Gomez Gonzalez et al. 2017). The method uses the annulus-wise approach presented in Mawet et al. (2014). No source is detected above 5σ. describe the noise of our images. Our next task is to look for The green circle delineates the planet’s project separation at ;3.5 au. point sources, and if none are found, place meaningful upper limits. Whether or not point sources are found, we will use our statistics, the traditional τ=5σ threshold yields an FPF of data to constrain the planet mass posterior distribution as a 2.98×10−7. function of projected separation. τ= σ / fl Applying the 5 threshold to the S N map generated For this task, we choose to convert ux levels into mass from our most sensitive reduction, which occurs for a number of estimates using the COND evolutionary model (Baraffe et al. ) principal components equal to 120, yields no detection, 2003 for the three ages considered in this work: 200, 400, and consistent with a null result. In other words, ò Eridani b is not 800 Myr. The young age end of our bracket (200 Myr) is σ ( ) detected in our deep imaging data to the 5 threshold. To derived from a pure kinematic analysis Fuhrmann 2004 . The compute the S/N map, we used the annulus-wise approach 400 and 800 Myr estimates are from Mamajek & Hillenbrand ( ) ( ) outlined in Mawet et al. 2014 , and implemented in the open- 2008 , who used chromospheric activities and spin as age source Python-based Vortex Imaging Pipeline (Gomez Gonzalez indicators. ) ( λ/ ) ( ) et al. 2017 . The noise in an annulus at radius r units of D is As noted by Bowler 2016 , the COND model is part of the computed as the standard deviation of the n=2πr resolution hot-start model family, which begins with arbitrarily large radii fi elements at that radius. The algorithm throughput is computed and oversimpli ed, idealized initial conditions. It ignores the using fake companion injection-recovery tests at every location effects of accretion and mass assembly. The COND model in the image. This step is necessary to account for ADI self- represents the most luminous—and thus optimistic—outcome. ò subtraction effects. The result is shown in Figure 2. At the adolescent age range of Eridani, initial conditions of To quantify our sensitivity, also known as “completeness,” the formation of a Jupiter-mass gas giant have mostly been we use the true positive fraction (TPF),defined as forgotten (Marley et al. 2007; Fortney et al. 2008) and have a minor impact on mass estimates. Moreover, a very practical TP +¥ reason why COND was used is because it is the only model TPF = = ò pxHdx(∣1 ) ()2 readily providing open-source tables extending into the low- TP+ FN t mass regime 9<1 MJup) reached by our data (see Section 5.1). with pxH(∣1 ), the probability density function of x under the — 4.1.1. Direct Imaging Nondetection hypothesis H1 signal present, and where TP is the number of true positives and FN the number of false negatives. For Signal detection is a balancing act where one trades off the instance, a 95% sensitivity (or completeness) for a given signal risk of false alarm with sensitivity. The signal detection I and detection threshold τ means that 95% of the objects at the threshold τ is related to the risk of false alarm, or false positive intensity level I will statistically be recovered from the data. fraction (FPF), as follows: The sensitivity contours, or “performance maps” (Jensen- ) +¥ Clem et al. 2018 for a uniform threshold corresponding to FP × −7 FPF = = ò pxH(∣0 ) dx ()1 2.98 10 FPF are shown in Figure 4. The choice of TN+ FP t threshold is assuming Gaussian noise statistics and accounts for small sample statistics as in Mawet et al. (2014). At the location where x is the intensity of the residual speckles in our images, of the elusive RV exoplanet, the threshold corrected for small pxH(∣0 )is the probability density function of x under the null sample statistics converges to τ≈5σ. The corresponding hypothesis H0, FP is the number of false positives, and TN is traditional τ=5σ contrast curve at 50% completeness is the number of true negatives. Assuming Gaussian noise shown in Figure 3.
5 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al. mass sensitivities using the hot start COND evolutionary model are ;3 MJ, ;4.5 MJ, and ;6.5 MJ at 1 au for all three ages considered here, i.e., 200, 400, and 800 Myr, respectively; at 2 au, they are ;1.5 MJ, ;1.7 MJ, and ;2.5 MJ, respectively; at 3 au, they are ;0.8 MJ, ;1.7 MJ, and ;5 MJ, respectively. These results demonstrate the power of ground-based Ms-band small-angle coronagraphic imaging for nearby adoles- cent systems. When giant exoplanets cool down to below 1000 K, the peak of their blackbody emission shifts to 3–5 μm mid-infrared wavelengths. Moreover, due to the t−5/4 depend- ence of bolometric luminosity on age (Stevenson 1991), mid- infrared luminosity stays relatively constant for hundreds of millions of years.
4.2. Tests on the RV Data In light of our nondetection of a planet in the NIRC2 high- contrast imaging, we consider the possibilities that the planet is not real or that the periodicity is caused by stellar activity. We utilize the RV analysis package RadVel26 (Fulton et al. 2018) to perform a series of tests to determine the significance of the periodicity and attempt to rule out stellar activity as its source. Figure 3. Traditional τ=5σ contrast curves comparing our Keck/NIRC2 We also test whether rotationally modulated noise must be vortex coronagraph Ms-band data to the VLT/NACO Lp-band data (PI: Quanz, considered in our analysis, and search for additional planets in Program ID: 090.C-0777(A)) presented in Mizuki et al. (2016) and reprocessed the RV data set. here with the VIP package.
4.1.2. Comparison to Previous Direct Imaging Results 4.2.1. Significance of the 7 yr Periodicity Mizuki et al. (2016) presented an extensive direct imaging First, we perform a one-planet fit to the RV data using compilation and data analysis for ò Eridani. The authors RadVel, and compare this model to the null hypothesis of no analyzed data from Subaru/HiCIAO, Gemini/NICI, and VLT/ Keplerian orbit using the Bayesian Information Criterion (BIC) NACO. Here, we focus on the deepest data set reported in to determine the significance of the 7 yr periodicity. The results Mizuki et al. (2016), which is the Lp-band NACO data from PI: of the RadVel MCMC analysis are located in Table 3, where ( ( )) T Quanz Program ID: 090.C-0777 A . This non-coronagraphic Pb is the planetary orbital period, conjb is the time of ADI sequence totals 146.3 minutes of integration time and conjunction, eb is the planetary eccentricity, ωb is the argument about 67° of parallactic angle rotation. Mizuki et al. (2016) of periastron of the planet, and Kb is the Keplerian semi- report 5σ and 50% completeness mass sensitivities using the amplitude. γ terms refer to the zero-point RV offset for each hot start COND evolutionary model that are >10 MJ at 1 au for instrument, and σ terms are the jitter, added in quadrature to the all three ages considered here, i.e., 200, 400, and 800 Myr; at measurement uncertainties as described in Section 2.2. The 2 au, they are ;2.5 MJ, ;4 MJ, and ;6.5 MJ, respectively; at maximum likelihood solution from the RadVel fit is plotted in 3 au, they are ;2 MJ, ;3 MJ, and ;5 MJ, respectively. Figure 6 against the full RV data set. For consistency, we reprocessed the VLT/NACO data with For the fit, the orbit is parameterized with eb, ωb, Kb, Pb, and T (γ) (σ) the VIP package and computed completeness maps using the conjb, as well as RV offsets and jitter terms for each same standards as for our Keck/NIRC2 data. The results are instrument. Due to the periodic upgrades of the Lick/Hamilton shown in Figure 5. Our computed 5σ and 50% completeness instrument and dewar, we split the Fischer et al. (2014) Lick mass sensitivities using the hot start COND evolutionary model data into four data sets, each with its own γ and jitter σ for the VLT/NACO Lp-band data are >10 MJ at 1 au for all parameter. This is warranted because Fischer et al. (2014) three ages considered here, i.e., 200, 400, and 800 Myr; at 2 au, demonstrated that statistically significant offsets could be they are ;6.5 MJ at 200 Myr and >10 MJ at both 400 and measured across the four upgrades in time series data on 800 Myr; at 3 au, they are ;5.5 MJ, ;8 MJ, and >10 MJ, standard stars. The largest zero-point offset they measured was respectively. a13m s-1 offset between the third and fourth data set. Our computed 5σ and 50% completeness results for the Although these offsets should have been subtracted before the VLT/NACO Lp-band data are systematically worse than those Lick/Hamilton data were published, the relative shifts between presented in Mizuki et al. (2016). We note a discrepancy in our derived γ parameters match well with those reported in mass between the published results and our values, by a factor Fischer et al. (2014) for each upgrade, implying that the offsets of two. We suggest that it may be the result of inaccurate flux were not subtracted for ò Eridani. loss calibrations in Mizuki et al. (2016), which is a common We find that the best-fit period is 7.37±0.08 yr, and that occurrence with ADI data sets. this periodicity is indeed highly significant, with ΔBIC= We find that our Ms-band Keck/NIRC2 coronagraphic data 245.98 between the one-planet model and the null hypothesis is about a factor of 5–10 more sensitive in mass, across the of no planets. Additionally, a model with fixed zero eccentricity range of solar system scales probed in this work, than the previous best available data set. Our 5σ and 50% completeness 26 Documentation available athttp://radvel.readthedocs.io/en/latest/.
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Figure 4. Keck/NIRC2 Ms-band vortex performance/completeness maps for a τ=5σ detection threshold for all three different ages considered here. The red curve highlights the 95% completeness contour. is preferred (ΔBIC=8.5) over one with a modeled the significance of the planet periodicity. The 7 yr periodicity in eccentricity. the residuals is still clearly visible by eye, and a one-planet fit to the RV residuals after the activity cycle is subtracted yields a 4.2.2. Source of the 7 yr Periodicity ΔBIC=197.4 when compared to a model with no planet. We next checked the RVs for correlation with SHK. Minor We next assess whether it might be possible that the source correlation was detected for the Keck/HIRES data set, with a of the periodicity at 7.37 yr is due to stellar activity, rather than Spearman correlation coefficient of rS=0.28 at moderate a true planet. statistical significance (p=0.01). For the APF data set, a To probe the potential effects of the magnetic activity on the stronger and statistically significant correlation was found RV periodicities, we examined time series data of the RVs (rS=0.50, p=0.01) between SHK and RV. However, given along with the SHK values from Lick, Keck, and the APF. RV that the 3 yr magnetic cycle shows up in the RV residuals, it is and SHK time series and Lomb Scargle periodograms are not surprising that RV and SHK might be correlated. There is plotted in Figure 7. also rotationally modulated noise that might be present in both A clear periodicity of 7.32 yr dominates the periodogram of data sets near the ∼11 day rotation timescale, increasing the the RV data set. This is within the 1σ credible interval of the correlation. We attempt to determine whether the measured best-fit periodicity found with RadVel, which yielded a correlation derives from the 3 yr periodicity in both data sets, or ΔBIC>200 when we tested its significance. Once the best-fit whether it is produced by other equivalent periodicities in the Keplerian planetary orbit from RadVel is subtracted from the RV and S data sets. RV data, the residuals and their periodogram are plotted in HK To test this, we first performed a Keplerian fit with a period panels (2a) and (2b) of Figure 7. The peak periodicity observed of approximately 3 yr to the Keck and APF S time series in the periodogram of the RV residuals is located at HK using RadVel. Although stellar activity is not the same as approximately 3 yr, coincident with the periodicity of the S HK orbital motion, we used the Keplerian function as a proxy for time series. the long-term stellar activity cycle of ò Eridani. We found a For the SHK time series, we detect clear SHK periodicities +30 near 3 yr, indicative of a ∼3 yr magnetic activity cycle (panels maximum probability period of 1194-25 days for the HIRES data and 989+40 days for the APF data. These periodicities are (4)–(5b)). We note that the peak SHK periodicity appears to be -26 slightly discrepant between the Keck and APF data sets indeed discrepant by more than 5σ. When combined, we found +22 (PKeck=3.17 yr; PAPF=2.59 yr), but consistent within the a period of 1147-20 days for the full HIRES and APF data set. FWHM of the periodogram peaks. This discrepancy likely We then subtracted the maximum probability 3 yr fit from results from a variety of causes, including the shorter time each SHK data set, and examined the residual values. We found baseline of the APF data, which covers only a single S cycle, that the correlation between these SHK residuals and the RV HK fi = and the typically non-sinusoidal and quasiperiodic nature of data was signi cantly reduced for the APF data, with rS 0.17 stellar activity cycles. The data sets also show a small offset in and p=0.04. This suggests that the strong correlation we the median SHK value, likely due to differing calibrations detected was primarily a result of the 3 yr periodicity. For the between the instrumental and telescope setups. However, the HIRES data set, the moderately significant correlation of = amplitude of the SHK variations appears consistent between the rS 0.28 was unchanged. data sets. A Lomb–Scargle periodogram of the SHK residuals is We next test whether the 3 yr activity cycle could be displayed in Figure 8. It shows no significant peak or power responsible for contributing power to the 7 yr periodicity. The near the posterior planet period at 2691 days, demonstrating longer period is not an alias of the 3 yr activity cycle, nor is it in that the SHK time series has no significant periodicity at the a low-order integer ratio with the magnetic activity cycle. We planet’s orbital period. perform a Keplerian fit to the RV time series from HIRES Our next tests involved modifying our one-planet fits to the and the APF, with a period constrained at the stellar activity RV data to account for the stellar activity cycle in two ways. period (1147 days).Wefind an RV semi-amplitude of K = We first performed a one-planet fit to the RVs using a linear +2.2 -1 +0.24 fi 4.8-1.7 m s and a large eccentricity of 0.53-0.27 ts the data decorrelation against the SHK values for the HIRES and APF set best. We then subtract this fit from the RV data to determine data sets. We then performed a two-Keplerian fit to the RV whether removal of the activity-induced RV periodicity affects data, in order to simultaneously characterize both the planetary
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Figure 5. Performance/completeness maps for a τ=5σ detection threshold for all three different ages considered here using VLT/NACO Lp-band data (PI: Quanz, Program ID: 090.C-0777(A)) presented in Mizuki et al. (2016). The red curve highlights the 95% completeness contour.
Table 3 the second Keplerian are clearly not converged over the RadVel MCMC Posteriors iterations completed for this model. Increasing the number of Parameter Credible Interval Maximum Likelihood Units iterations does not appear to improve convergence. This again +29 points to the quasiperiodic nature of stellar activity cycles, and Pb 2691-28 2692 days S +800 the different time baselines of the full RV data set and the HK Tconjb 2530054-770 2530054 JD time series available. The fit has an RV semi-amplitude of +0.061 eb 0.071-0.049 0.062 K =4.4 m s-1, lower than the semi-amplitude of the ω +0.82 activity b 3.13-0.79 3.1 radians planet at K =11.81±0.65 m s-1. −1 b Kb 11.48±0.66 11.49 m s γ ± -1 HIRES 2.0 0.89 2.05 ms 4.2.3. Search for Additional RV Planets -1 γHARPS −4.6±1.6 −4.8 ms -1 We used the automated planet search algorithm described by γAPF −3±1 −3 ms g - +0.99 − ms-1 Howard & Fulton (2016) to determine whether additional Lick4 2.45-0.96 2.41 g 10.5+2.0 11.0 ms-1 planet signatures are present in the combined RV data set. The Lick3 -1.9 fi g 8.6±2.3 8.6 ms-1 residuals to the two-Keplerian t were examined for additional Lick2 periodic signatures. The search was performed using a 2D g 8.5±2.5 8.5 ms-1 Lick1 Keplerian Lomb–Scargle periodogram (2DKLS)(O’Toole γ ± -1 CES+LC 6.7 2.7 6.7 ms ) -1 et al. 2009 . γCES+VLC 3.2±1.8 3.3 ms − − The residuals to the two-Keplerian fit show several small g˙ ≡0.0 ≡0.0 m s 1 d 1 − − peaks, but none with empirical false alarm probabilities (eFAP) g¨ ≡0.0 ≡0.0 m s 1 d 2 (Howard & Fulton 2016) less than 1% (Figure 7, panel (3b)).A σ 5.99+0.88 5.83 ms-1 HIRES -0.82 broad forest of peaks at approximately 12 days corresponds to σ 5.3+1.7 4.8 ms-1 HARPS -1.6 the stellar rotation period and is likely due to spot-modulated σ +0.75 5.12 ms-1 APF 5.26-0.72 stellar jitter. The next most significant peak is located at 108.3 s 7.1+0.96 6.85 ms-1 Lick4 -0.88 days. We attempted a three-Keplerian fit to the RVs with the s 7.6+1.7 7.2 ms-1 Lick3 -1.5 third Keplerian initiated at 108.3 days. However, we were s 2.8+2.3 0.0 ms-1 Lick2 -1.6 unable to achieve convergence in a reasonable number of s 14.5+2.4 13.9 ms-1 Lick1 -2.1 iterations, and the walkers were poorly behaved. This serves as +3.0 -1 sCES+ LC 9.8-2.7 9.0 ms evidence against the inclusion of a third periodicity. We σ +2.2 -1 CES+VLC 4.4-2.1 3.6 ms conclude that there is insufficient evidence to suggest an inner planet to ò Eridani b exists. Note. 860,000 links saved. RV and residual time series, as well as 2DKLS periodograms Reference epoch for gg, ˙ , g¨ : 2452438.84422. used for the additional planet search, are plotted in Figure 7, panel (3a) and (3b). orbit and the stellar activity cycle. In both cases, we checked for significant changes to the planetary orbital parameters due 4.2.4. Gaussian Processes Fits to accounting for the stellar activity cycle in the fit. For both For all of these analyses, we have assumed white noise and tests, we find that the maximum likelihood values of the “ ” ’ σ added a jitter term in quadrature to account for uncertainty planet s orbital parameters all agree within 1 credible intervals due to stellar activity. We assessed whether this was reasonable fi with the single-planet t. by performing a one-planet fit using RadVel and including a fi fi We note that, from the two-Keplerian t, the best- t second Gaussian processes model to account for rotationally modu- Keplerian provides some information about the stellar activity lated stellar noise (S. Blunt & A. W. Howard 2018, in cycle. It has a best-fit period of 1079 days, or 2.95 yr, shorter preparation; M. Kosiarek et al. 2018, in preparation) as well as than the periodicity derived from a fit to the HIRES and APF the 3 yr stellar activity cycle. SHK time series. However, though the other parameters in this First, we used RadVel with a new implementation of GP fit seem to be converged, the period and time of conjunction of regression using the quasiperiodic covariance kernel to fit four
8 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al.
Figure 6. Time series and phase-folded radial velocity curves from all data sets are plotted. The maximum probability single-Keplerian model from RadVel is overplotted, as are the binned data (red). The plotted error bars include the internal rms derived from the RV code, as well as the fitted stellar and instrumental jitter parameter σj for each instrument.
GP hyperparameters in addition to the Keplerian parameters for to correlated noise on the rotation timescale, resulting in GP a single planet and a white noise term σj. The hyperparameters amplitudes consistent with zero. For other instruments, notably the for the quasiperiodic kernel are the amplitude of the covariance HIRES and APF data, the white noise jitter term σj was function (h); the period of the correlated noise (θ, in this case significantly reduced in the GP model, compared with the standard trained on the rotation period of the star); the characteristic RV solution. decay timescale of the correlation (λ, a proxy for the typical However, when comparing the derived properties of the spot lifetime); and the coherence scale (w, sometimes called the planet, we find that the GP analysis has no noticeable effect on ’ structure parameter)(Grunblatt et al. 2015; López-Morales the planet s orbital parameters. The period, RV semi- et al. 2016). amplitude, eccentricity, time of conjunction, and argument of We applied a Gaussian prior to the rotation period of periastron constraints from the GP regression analysis all agree σ θ=11.45±2.0 days, based on the periodicity observed in the within 1 with the values derived from the traditional one- fi fi planet fit. We therefore conclude that the rotationally RV residuals to the two-Keplerian t, but suf ciently wide to fi ’ allow the model flexibility. The covariance amplitudes h for modulated noise does not signi cantly affect the planet s orbital parameters. each instrument were constrained with a Jeffrey’s prior We additionally performed a one-planet fit using GP truncated at 0.1 and 100 m s-1. We imposed a uniform prior regression to model the 3 yr stellar activity cycle. For this test of 0–1 yr on the exponential decay timescale parameter λ.We case, we used a periodic GP kernel because each data set covers chose a Gaussian prior for w of 0.5±0.05, following López- ( ) only a relatively few cycles of the stellar activity cycle. Unlike Morales et al. 2016 . activity signatures at the stellar rotation period, we do not The results of our GP analysis provide constraints on the hyp- fi +0.33 expect to see signi cant decay or decorrelation of the 3 yr cycle erparameters, indicating that the rotation period is 11.64-0.24 days over the time span of our data set. This periodic GP model had +15 and the exponential decay timescale is 49-11 days. The amplitude hyperparameters describing the periodicity (θ), amplitude (h), parameters for each instrument ranged from 0.0 to 13.4 m s-1,and and structure parameter (w), but no exponential decay. This were highest for the earliest Lick RV data. For some of the data analysis is somewhat akin to our two-Keplerian fit, but allows sets, the cadence of the observations likely reduced their sensitivity more flexibility to fit the noise than a Keplerian. For this model,
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Figure 7. Time series (a) and Lomb–Scargle periodograms (b). Panels (1)–(3) show the periodicities of the radial velocity measurements, residuals to a one-Keplerian (planet) RadVel fit, and residuals to a two-Keplerian (planet + stellar activity) RadVel fit, respectively. Panels (4) and (5) show the time series and periodograms of the Keck and APF SHK values. The peak periodicities for each data set are indicated in the periodogram plots. The periodicities of the SHK data sets (panels 4–5) are overplotted in the second periodogram panel (2b), showing the correspondence between SHK periodicity and the secondary, activity-induced peak in the RV residuals. The broad, low-significance peak at 11.45 days in panel (3b) corresponds to the stellar rotation period. Plotting symbols for the RV data sets are the same as in Figure 6. we placed a Gaussian prior of θ=1147±20 days on the GP The results of this analysis indicate a GP periodicity of period parameter, based on the Keplerian fittotheSHK values. θ=1149±17 days, a slightly tighter constraint than the We found that, when allowing each instrument its own GP imposed prior. The GP amplitude parameter was constrained to +1.21 -1 amplitude parameter, h, nearly all of the instrumental amplitudes be h = 4.26-1.01 m s , comparable with the posteriors for the were best fit with values very close to zero, so instead we fitfor RV semi-amplitude of the second Keplerian in the two- only a single GP amplitude across all instrumental data sets. We Keplerian fit. Importantly, the model posteriors on the constrained this parameter with a Jeffreys prior bounded at planetary parameters were again consistent within 1σ with 0.01–100 m s-1. We again used the w=0.5±0.05 prior for the our traditional one-planet fit for all parameters, including the fi structure parameter, after testing out ts at several values white noise jitter terms σj as well as the orbital period, RV between 0 and 1. It remains unclear whether this was the optimal semi-amplitude, eccentricity, and Keplerian angles. choice, given that the physical interpretation of this parameter We note that the traditional one-planet fit is preferred over would be different for the long-term stellar activity cycle as the one-planet Gaussian processes fitbyΔBIC=19.7. The compared with the rotationally modulated spot noise. two-Keplerian fit is also preferred over the GP one-planet
10 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al. noise is uncorrelated and that Σ is therefore diagonal dimn=+ ()4 with m=m(x) being a normalized planet model at the position x. Assuming Gaussian noise, the direct imaging likelihood is given by: 1 1 (∣di, x )=---exp ()()dm iS-1 dm i 2p∣∣S {}2 1 µ-exp ()ii21mmdmSS-- -2. 1 {}2 ()5
Figure 8. Periodogram of the HIRES and APF SHK residuals to the ∼3 year fit. -1 The red dotted line shows the best-fit period of the planet from our initial one- We have used the fact that d S d is a constant because we are planet fit. Like the SHK periodograms shown in Figure 7 panels (4)–(5b), there not inferring the direct imaging covariance. ’ is no power at the planet s orbital period. Even when the peak periodicity is The estimated brightness i˜x, in a maximum likelihood sense, removed for each data set, no additional power appears at the planet’s 7.37 year and associated error bar σx are defined as: orbital period. This indicates that stellar activity is not likely to cause the 7.37 year periodicity in the radial velocity data. dmS-1 i˜x = ,6() mmS-1 model, with ΔBIC=30.6, despite having two additional free parameters and nominally less flexibility than the GP model. and These tests demonstrate that the addition of a Keplerian or 2 --11 s x = ()mmS .7 () Gaussian-process model to account for stellar activity (both rotationally modulated activity and the long-period stellar We can therefore rewrite the logarithm of the direct imaging activity cycle) does not strongly influence the results of the likelihood as a function of these quantities (Ruffio et al. 2018), fi fi planetary orbital t. The GP t in particular was statistically 1 (∣di x )=- ( i2 - ii˜ )() disfavored compared to the simpler Keplerian model based on log , 2 2.x 8 Δ BIC. We therefore choose to restrict our subsequent analyses 2s x to consider only a single planet and only white noise. Going The definition of the planet model m is challenging when forward, the uncertainty due to stellar activity is added in using a PCA-based image processing. Indeed, while it subtracts quadrature as a white-noise “jitter” term and red noise is not the speckle pattern, it also distorts the signal of the planet. The considered. distortion is generally not accounted for in a classical data reduction such as the one used in Section 4.1, which is why it is more convenient to adopt a Forward Model Matched Filter 4.3. Combining Constraints from Imaging and RV (FMMF) approach as described in Ruffio et al. (2017). The By combining the imaging and RV data sets, it is possible to FMMF computes the map of estimated brightness and standard place tighter upper limits on the mass of the companion. deviation used in Equation (8) by deriving a linear approx- Indeed, the RV data provides a lower limit on the planet mass imation of the distorted planet signal for each independent (M sin i), while the direct imaging data complements it with an exposure, called the forward model (Pueyo 2016). upper limit. We showed that the likelihood can theoretically be An MCMC will be used to infer the posterior on the masses calculated directly from the final products of the FMMF. In and orbital parameters of the system, noted as Θ. The noise in practice, the noise is correlated and not perfectly Gaussian, the RV measurements dRV and in the images dDI is resulting in the standard deviation being underestimated and independent, which means that the joint likelihood is thus possibly biasing the estimated brightness. We therefore separable: recalibrate the S/N by dividing it by its standard deviation computed in concentric annuli. The estimated brightness map is (∣)(∣)(∣)()ddDI,.3 RVQ= d RV Q d DI Q corrected for algorithm throughput using simulated planet injection and recovery. The likelihood is computed for the fully calibrated S/N maps. FMMF is part of a Python implementation of the PCA algorithm presented in Soummer et al. (2012) called PyKLIP27 4.3.1. Direct Imaging Likelihood (Wang et al. 2015). The principal components for each In this section, we detail the computation of the direct exposure are calculated from a reference library of the 200 imaging likelihood (Ruffio et al. 2018). The direct imaging data most correlated images from which only the first 20 modes are dDI, temporarily shortened to d, is a vector of Nexp×Npix kept. Images in which the planet would be overlapping with the elements where Nexp is the number of exposures in the data set current exposure are not considered to be part of the reference and Npix the number of pixels in an image. It is the library, to limit the self- and over-subtraction using an concatenation of all the vectorized speckle subtracted single exclusion criterion of seven pixels (0.7λ/D). The speckle exposures. A point source is defined from its position x and its fi 27 brightness i. We also de ne n as a Gaussian random vector Available under open-source license athttps://bitbucket.org/pyKLIP/ with zero mean and covariance matrix Σ. We assume that the pyklip.
11 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al. subtraction is independently performed on small sectors of the image.
4.3.2. Joint Likelihood and Priors We implement a Markov-chain Monte Carlo analysis of the combined RV data from the Coudé Echelle Spectrograph, HARPS, Lick/Hamilton, Keck/HIRES, and APF/Levy instru- ments, as well as the single-epoch direct imaging data. We solve for the full Keplerian orbital parameters, including orbital inclination and longitude of the ascending node, which are not typically included in RV-only orbital analyses. Including the full Keplerian parameters allows us to calculate the projected position of the companion at the imaging epoch for each model orbit. This is necessary to calculate an additional likelihood based on the direct imaging data. The full log-likelihood function used for this analysis is:
1 (∣)(ddQ=- i2 - ii˜ ) logDI , RV 2 2 x 2s x ⎡ ⎤ (())vvt- 2 - ⎢ imi++log 2ps()()22 s ⎥ . 9 å⎢ 22 ij⎥ i ⎣ 2()ssij+ ⎦ Figure 9. Corner plot showing the posterior distributions and correlation between the companion mass and inclination for models using the RV The RV component of the likelihood comes from (Howard likelihood only, as well as RV + direct imaging likelihood with planet models et al. 2014). Here, vi=vi,inst−γinst is the offset-subtracted RV of age 800, 400, and 200 Myr (a log-uniform prior). measurement; σi refers to the internal uncertainty for each measurement; vm(ti) is the Keplerian model velocity at the time that average acceptance fractions for our chains are fairly low, of each observation; and σj is the instrument-specific jitter term, ≈5%–10%. which contributes additional uncertainty due to both stellar activity and instrumental noise. In these models, each 4.3.3. MCMC Results instrument’s RV offset (γinst) and jitter term (σj,inst) are included as free parameters in the fit. A description of the The planet parameters derived in this analysis are consistent direct imaging component of the likelihood is available in with those determined by RadVel. The posterior distributions Section 4.3.1. for the companion mass and orbital inclination are plotted in We draw from uniform distributions in log P, log Mb, cos i, Figure 9. The lower limit on planet mass Mib sin= 0.72 Ω fi e cos w, e sin w, , mean anomaly at the epoch of the rst 0.07 MJup is constrained by the Keplerian velocity semi- observation, and γinst. amplitude and agrees well with the RadVel results. With the We place a tight Gaussian prior of Må=0.781±0.078 Me RV data alone, the true mass (independent of sin i) has a poorly on the primary stellar mass, based on the interferometric results constrained upper limit, although high-mass, low-inclination of Boyajian et al. (2012). Other groups have measured slightly orbits are geometrically disfavored. With the addition of the different but generally consistent stellar masses for ò Eridani. imaging nondetection constraints, the mass upper limit is Valenti & Fischer (2005) report a spectroscopic mass of improved. Må=0.708±0.067 Me; Takeda et al. (2007) report a Because younger planets are hotter and thus brighter, the +0.06 discrepant spectroscopic result of MM = 0.856-0.08 . direct imaging likelihood disfavors a broader region of A tight Gaussian prior of π=310.94±0.16 mas is also parameter space when a younger age is assumed. Thus, the imposed on stellar parallax based on the Hipparcos parallax tightest constraints come from the youngest-aged planet measurement for this star (van Leeuwen 2007). We place wide models. Table 4 lists the planet parameters resulting from each -1 Gaussian priors on the jitter terms, with σj=10.0±10.0 m s . MCMC run. We report the median and 68% credible intervals Large values for jitter are also disfavored by the second term of for each model. the likelihood function. We also calculate the posterior distribution on the position of With these priors and this likelihood function, we solve for the planet at the epoch of the NIRC2 imaging observation from the full orbital parameters and uncertainties using the Python the RV-only likelihood model. We check this posterior to package emcee (Foreman-Mackey et al. 2013). For compar- ensure that the imaging observations were optimally timed to ison with the RadVel results, we perform our analysis both detect the planet at maximal separation from the star. The with and without the direct imaging likelihood. We use planet positional posterior distribution is plotted in Figure 11;it models of ages 800, 400, and 200 Myr in individual analyses, demonstrates that, at the epoch of the imaging observations, the because the system’s age constraints span this range. We use separation of the planet from the star was indeed maximized. the standard emcee Ensemble Sampler; each MCMC run uses The planet would have been easily resolvable, regardless of the 100 walkers and is iterated for more than 500,000 steps per on-sky orientation (i.e., the longitude of the ascending node). walker. We check that each sampler satisfies a threshold of For these analyses, we draw companion mass uniformly in Gelman–Rubin statistic Rˆ < 1.1 for all parameters (Gelman & logarithmic space with bounds at 0.01 and 100 MJup. This is Rubin 1992; Ford 2006), to test for nonconvergence. We note comparable to placing a Jeffreys prior—a common choice of
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Table 4 MCMC Results
RV Likelihood Only 800 Myr 400 Myr 200 Myr Parameter Median and 68% Credible Interval ( ) +0.43 +0.38 +0.19 +0.09 mb MJup 0.78-0.12 0.78-0.12 0.75-0.10 0.71-0.07 ( ) +0.07 +0.07 +0.07 +0.07 P yr 7.37-0.07 7.37-0.07 7.37-0.07 7.38-0.07 +0.06 +0.06 +0.06 +0.06 e 0.07-0.05 0.07-0.05 0.07-0.05 0.06-0.04 ω (°) +49 +53 +48 +66 177-51 175-52 177-49 157-51 Ω (°) +122 +126 +108 +47 180-123 184-131 212-148 276-158 (°) +42 +42 +35 +23 i 90-43 89-42 89-35 90-24 ( ) +336 +361 +332 +475 tperi JD 2447213-429 2447198-426 2447218-407 2447032-402
γ ( -1) +2.4 +2.4 +2.4 +2.4 Lick1 ms 8.4-2.4 8.5-2.4 8.4-2.3 8.4-2.4 σ ( -1) +2.0 +2.0 +2.1 +2.1 Lick1 ms 15.1-1.8 15.1-1.8 15.1-1.8 15.1-1.8 γ ( -1) +2.6 +2.6 +2.6 +2.6 CES+LC ms 6.9-2.6 6.9-2.6 6.9-2.6 6.9-2.5 σ ( -1) +2.5 +2.5 +2.5 +2.5 CES+LC ms 10.9-2.1 10.9-2.1 10.9-2.2 10.9-2.2 γ ( -1) +1.9 +1.9 +1.9 +2.0 Lick2 ms 8.5-1.9 8.5-1.9 8.5-1.9 8.5-1.9 σ ( -1) +2.1 +2.2 +2.1 +2.2 Lick2 ms 4.7-1.3 4.7-1.4 4.7-1.4 4.8-1.4 γ ( -1) +1.9 +1.8 +1.9 +1.9 Lick3 ms 10.5-1.9 10.6-1.9 10.5-1.9 10.6-1.8 σ ( -1) +1.4 +1.4 +1.4 +1.4 Lick3 ms 9.2-1.2 9.2-1.1 9.2-1.1 9.2-1.1 γ ( -1) +1.7 +1.8 +1.8 +1.8 CES+VLC ms 3.4-1.8 3.4-1.9 3.4-1.8 3.4-1.8 σ ( -1) +1.8 +1.8 +1.7 +1.8 CES+VLC ms 6.8-1.5 6.8-1.5 6.9-1.5 6.8-1.5 γ ( -1) +1.0 +1.0 +1.0 +1.0 Lick4 ms -2.4-1.0 -2.4-1.0 -2.4-1.0 -2.4-1.0 σ ( -1) +0.8 +0.7 +0.7 +0.7 Lick4 ms 8.7-0.7 8.7-0.7 8.7-0.7 8.7-0.7 γ ( -1) +1.6 +1.5 +1.5 +1.6 HARPS ms -4.5-1.6 -4.5-1.5 -4.5-1.5 -4.4-1.6 σ ( -1) +1.3 +1.3 +1.3 +1.3 HARPS ms 7.4-1.0 7.4-1.0 7.4-1.0 7.4-1.0 γ ( -1) +0.9 +0.9 +0.9 +0.9 HIRES ms 2.0-0.9 2.0-0.8 2.0-0.9 1.9-0.9 σ ( -1) +0.7 +0.7 +0.7 +0.6 HIRES ms 7.9-0.6 7.8-0.6 7.9-0.6 7.8-0.6 γ ( -1) +1.0 +1.0 +1.0 +1.0 APF ms -2.7-1.0 -2.7-1.0 -2.7-1.0 -2.7-1.0 σ ( -1) +0.5 +0.5 +0.5 +0.5 APF ms 7.3-0.5 7.3-0.5 7.3-0.5 7.3-0.5 prior for scale parameters such as mass and period (Ford 2006). of detecting additional planets with future facilities such as the This prior is also not significantly dissimilar to the mass James Webb Space Telescope. distribution of Doppler-detected Jovian planets from Cumming et al. (2008), who found that dN µ M-0.31, a roughly flat 5.1. Choice of Evolutionary Models dmlog distribution in log m. The direct imaging upper limits are model-dependent. We To assess the impact of this choice, we repeat our analysis chose to use the COND model mostly for practical reasons. with a uniform prior on the mass, again from 0.01 to 100 MJup. This choice was also motivated by the fact that, at the system’s fi This alternative increases the signi cance of the tail of the mb age and probable planet mass, evolutionary models have posterior distribution toward higher masses. Because mass and mostly forgotten initial conditions such that hot and cold start inclination are highly correlated, this effect also serves to models have converged (Marley et al. 2007). However, COND fl fi atten out the inclination posterior, adding more signi cance to is arguably one of the oldest evolutionary models available. lower-inclination orbits. Figure 10 shows the posteriors and The treatment of opacities, chemistry, etc., are all somewhat correlation between the mass and inclination of the planet outdated. For our 800 Myr case, the most probable age for the fi under the modi ed mass prior. The correlation plot is identical system, we also generated completeness maps using the to that shown in Figure 9, and the mass posterior is not evolutionary model presented in Spiegel & Burrows (2012), / fi qualitatively changed. The median 68% con dence interval referred to as SB12 hereafter (Figure 12). Because the publicly +0.47 planet mass from the 800 Myr model is mMb = 0.83-0.15 Jup, available SB12 grid does not fully cover our age and mass consistent within uncertainties with the mass constraint from range, some minor extrapolations were necessary. The result of the log-mass case at the same age. The inclination posterior has this comparison shows some noticeable discrepancies across a wider uncertainty in the linear mass case (i=90°.8±48°.0) the range probed by our data (see Figure 12). However, both as compared to the log mass case (i=89°.2±41°.7). All other models seem to agree to within error bars at the location of the orbital and instrumental parameters have equivalent constraints planet around 3.48 au, so the impact of the choice of in both cases. We conclude that the prior on mass does not evolutionary model on our joint statistical analysis is only significantly affect the results of the analysis. marginal.
5. Discussion 5.2. Constraints on the System’s Age and Inclination In this section, we discuss the impact of our joint RV-direct The planet is not detected in our deep imaging data to the 5σ imaging analysis on the probable age of the system and the threshold. According to our upper limits and RV results, the possible planet–disk interactions. We also discuss the prospect imaging nondetection indicates that the true age of ò Eridani is
13 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al.
Table 5 Table 5 Keck Radial Velocity Measurements (Continued)
( −1)a σ ( −1)b − − JD RV ms RV ms S ( 1)a σ ( 1)b HK JD RV ms RV ms SHK − 2455110.97985 6.54 1.30 0.467 2457245.14532 5.26 0.90 0.479 − 2455171.90825 3.33 1.09 0.486 2457246.14242 −1.45 0.99 0.477 2455188.78841 7.90 1.11 0.481 2457247.14678 −5.60 1.01 0.482 − 2455231.7593 8.39 1.13 0.497 2457254.14889 8.50 0.80 0.475 2455255.70841 1.66 0.70 0.520 2457255.15244 6.36 0.91 0.466 2455260.71231 1.77 1.01 0.523 2457256.15168 5.80 0.83 0.476 2455261.71825 0.75 1.30 0.526 2457265.14924 5.74 0.88 0.469 − 2455413.14376 10.67 0.76 0.500 2457291.04683 6.07 1.05 0.491 − 2455414.13849 16.73 0.99 0.000 2457326.9831 6.10 1.12 0.501 − 2455415.14082 20.89 0.78 0.495 2457353.88153 −0.55 1.09 0.519 − 2455426.14477 17.57 0.86 0.494 2457378.78993 2.19 1.08 0.519 − 2455427.14813 18.05 0.87 0.483 2457384.78144 14.17 1.10 0.517 − 2455428.14758 21.46 0.87 0.480 2457401.75106 6.07 0.99 0.517 − 2455429.14896 18.67 0.90 0.475 2457669.02614 1.91 1.10 0.497 2455434.14805 7.21 0.86 0.474 2457672.99494 −1.33 1.20 0.497 2455435.14705 4.46 0.89 0.481 2457678.97973 −13.88 1.10 0.495 − 2455436.14535 2.48 0.83 0.485 2457704.03411 −14.12 0.67 0.501 − 2455437.15006 5.03 0.94 0.480 2457712.99284 −4.84 1.18 0.478 − 2455438.15172 14.24 0.90 0.484 2457789.74988 −13.12 1.12 0.439 − 2455439.14979 13.17 0.51 0.474 2457790.737 −8.09 1.01 0.440 − 2455440.15188 22.38 0.88 0.471 2457803.70407 −4.25 1.09 0.460 − 2455441.15033 19.71 0.99 0.469 2457804.70718 −6.55 1.09 0.471 2455456.01632 4.52 0.97 0.466 2457806.79201 −11.62 1.13 0.464 − 2455465.07401 12.99 0.98 0.449 2457828.7545 −12.69 1.12 0.455 2455469.1284 7.81 1.01 0.465 2457829.71875 −19.82 0.98 0.466 − 2455471.97444 4.15 1.16 0.471 2457830.71979 −12.66 1.10 0.465 2455487.00413 −9.44 0.96 0.454 − 2455500.98687 2.23 1.05 0.461 Notes. − 2455521.89317 11.42 1.05 0.455 a The RV data points listed in these tables are not offset-subtracted. − 2455542.95125 8.56 1.20 0.458 b Uncertainties quoted in these tables reflect the internal statistical variance of 2455613.70363 0.65 1.01 0.466 the spectral chunks used to extract the RV data points (see Section 2.2 for a full 2455791.13884 1.87 0.87 0.433 description). They do not include jitter. 2455792.13464 −9.19 0.90 0.430 ( ) 2455793.13858 −17.85 0.89 0.426 This table is available in machine-readable form. 2455795.14053 −15.43 0.96 0.418 2455797.13828 −5.67 0.83 0.419 2455798.14195 −5.00 0.84 0.424 likely to be closer to 800 Myr. Moreover, spectroscopic 2455807.1116 −3.91 0.99 0.417 indicators of age (log R¢ and rotation) point toward this star − HK 2455809.1367 0.90 0.99 0.429 being at the older end of the age range tested here, nearer to 2455870.9902 1.81 1.20 0.437 ( ) 2455902.82961 4.20 0.74 0.429 800 Myr than 200 Myr Mamajek & Hillenbrand 2008 . VLTI 2455960.69933 −8.22 1.21 0.460 observations were used to interferometrically measure the 2456138.12976 −2.69 0.86 0.464 stellar radius and place the star on isochrone tracks. These 2456149.05961 −2.49 0.53 0.470 models also yield an age of 800 Myr or more (Di Folco et al. 2456173.13157 −1.22 0.96 0.459 2004). Thus, the most likely model included here is the 2456202.99824 19.64 0.71 0.507 800 Myr model, which is also a fortiori the least restrictive in 2456327.70174 20.33 1.05 0.535 placing an upper limit on the planet mass. This model yields a 2456343.7026 16.52 1.05 0.505 +0.38 mass estimate of MMb = 0.78-0.12 Jup and an orbital plane 2456530.11763 6.76 0.90 0.489 inclination of i=89°±42°. 2456532.12218 8.06 0.85 0.479 2456587.96668 14.41 1.03 0.479 We note that this inclination is marginally consistent with 2456613.91026 15.04 1.02 0.481 being co-planar with the outer disk belt, which has a measured 2456637.81493 23.88 1.02 0.487 inclination of i=34°±2° (Booth et al. 2017). Although the 2456638.79118 32.35 1.07 0.491 direct imaging nondetection naturally favors near edge-on 2456674.80603 11.70 1.03 0.488 solutions, the full posterior distribution can still be interpreted 2456708.78257 2.49 0.99 0.482 as consistent with the planet being co-planar with the outer 2456884.13093 12.85 0.95 0.446 disk. With the joint RV and imaging analysis, we are unable to 2456889.14678 18.51 0.82 0.466 definitively state whether the planet is or is not co-planar with 2456890.14703 13.09 0.86 0.461 the outer debris disk. However, we are able to rule out ages at 2456894.13998 8.71 0.83 0.446 2456896.11131 15.09 0.78 0.447 or below 200 Myr if coplanarity is required. 2456910.94964 13.84 0.64 0.450 To assess the properties of the planet assuming coplanarity 2457234.13834 9.97 0.85 0.491 with the disk, we repeat our joint analysis, implementing a new 2457240.99109 6.26 0.52 0.468 Gaussian prior on the inclination of i=34°±2° rather than 2457243.14297 3.19 0.78 0.476 the uninformative geometric prior used in the previous analysis.
14 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al.
Table 6 Table 6 APF Radial Velocity Measurements (Continued)
( −1)a σ ( −1)b − − JD RV ms RV ms S ( 1)a σ ( 1)b HK JD RV ms RV ms SHK 2456582.93034 26.64 2.73 0.524 2457364.77113 −7.80 1.30 0.531 2456597.91368 6.40 2.36 0.528 2457365.70544 0.72 1.26 0.550 2456606.68427 16.52 0.75 0.531 2457424.71436 −1.68 1.37 0.555 2456608.10376 4.69 0.78 0.530 2457426.63205 3.62 1.42 0.559 2456610.7625 16.04 1.18 0.512 2457427.38923 3.97 1.17 0.577 − 2456618.88476 2.11 0.78 0.530 2457429.72793 2.42 0.90 0.560 2456624.72004 4.20 1.11 0.519 2457432.60322 6.20 1.25 0.569 2456626.81421 24.46 0.75 0.521 2457435.69406 −18.61 18.79 0.304 2456628.72976 24.14 0.70 0.540 2457443.66061 2.25 1.24 0.559 − 2456631.42746 2.26 0.88 0.502 2457446.70278 3.96 1.37 0.566 2456632.80921 14.46 0.62 0.523 2457471.55712 5.85 1.63 0.535 2456644.75696 8.20 2.30 0.522 2457599.93545 −5.69 0.85 0.505 2456647.81171 14.44 0.63 0.535 2457605.99828 −5.33 1.27 0.559 2456648.59184 12.62 1.10 0.538 2457607.92844 −24.97 1.39 0.540 2456662.63738 9.77 0.73 0.536 2457611.16197 −16.02 1.26 0.510 2456663.75415 10.43 1.11 0.531 2457613.86777 2.47 1.54 0.560 2456667.52792 18.00 0.78 0.535 2457615.04307 3.50 1.48 0.538 2456671.68695 19.96 1.05 0.604 2457617.08138 0.91 1.29 0.555 2456675.75647 7.84 1.12 0.519 2457619.05397 −12.30 1.46 0.529 2456679.83732 17.70 1.05 0.529 2457621.79772 −13.43 1.57 0.508 2456682.56608 17.80 0.82 0.550 2457626.10874 0.39 1.33 0.534 2456689.76638 26.34 0.75 0.500 2457627.95628 −4.92 1.37 0.551 2456875.02028 7.12 2.18 0.501 2457633.96762 −8.24 1.70 0.512 2456894.88054 8.28 1.30 0.470 2457636.08672 −1.33 1.18 0.539 2456901.06193 9.95 1.54 0.479 2457637.95848 −7.66 1.37 0.538 − 2456909.10279 4.71 1.21 0.476 2457643.92459 −14.39 1.33 0.512 2456922.07953 12.25 2.13 0.461 2457668.93315 −0.83 1.34 0.527 − 2456935.94021 2.43 1.27 0.479 2457669.90475 2.76 1.43 0.533 − 2456937.92403 0.55 1.35 0.468 2457670.88203 −8.82 1.42 0.543 2456950.03798 3.82 1.44 0.472 2457674.61398 −5.61 1.42 0.534 − 2456985.64755 1.80 2.28 0.441 2457679.98028 −12.42 1.78 0.515 2456988.63095 5.93 1.29 0.478 2457687.77138 1.17 1.37 0.524 2456999.76434 8.84 1.37 0.459 2457694.76122 −3.81 1.33 0.504 − 2457015.72916 2.17 1.10 0.465 2457696.82099 −5.60 1.32 0.522 − 2457026.78021 1.44 1.34 0.464 2457700.96748 −10.84 1.41 0.534 − 2457058.45996 3.69 1.89 0.435 2457701.84849 −11.69 1.38 0.517 2457234.08236 7.73 1.39 0.525 2457702.89789 −14.82 1.22 0.524 − 2457245.86234 4.19 1.41 0.519 2457703.82658 −19.89 1.25 0.523 − 2457249.93007 3.94 1.31 0.500 2457705.73282 −9.58 1.32 0.513 2457253.11257 5.63 1.33 0.511 2457707.78376 −9.03 1.24 0.511 − 2457257.15719 1.02 1.15 0.506 2457717.79818 −15.06 1.22 0.505 − 2457258.94437 12.69 1.23 0.517 2457722.75749 −12.43 2.05 0.427 − 2457261.02221 2.76 1.32 0.501 2457728.81592 −7.64 1.67 0.514 − 2457262.94505 7.81 1.36 0.496 2457741.79955 −14.52 1.16 0.513 2457265.95783 9.67 1.24 0.516 2457743.5028 −17.28 1.32 0.489 − 2457275.01304 1.91 1.23 0.515 2457745.93451 −17.74 1.31 0.487 2457283.96368 1.88 1.29 0.507 2457749.71344 −5.63 1.30 0.503 − 2457287.02735 1.11 1.35 0.524 2457751.64976 −16.16 1.32 0.501 2457290.95635 3.19 1.42 0.534 2457753.47716 −12.45 1.30 0.509 − 2457305.83659 5.63 1.23 0.515 2457798.55461 −18.91 2.25 0.465 2457308.90844 13.30 1.26 0.534 2457821.65582 −5.60 1.63 0.490 2457318.83435 8.72 1.26 0.557 2457321.79157 6.64 1.36 0.540 Notes. 2457325.84352 2.87 1.41 0.543 a The RV data points listed in these tables are not offset-subtracted. 2457331.10764 9.90 1.36 0.552 b Uncertainties quoted in these tables reflect the internal statistical variance of 2457332.78237 9.64 1.25 0.558 the spectral chunks used to extract the RV data points (see Section 2.2 for a full 2457334.82998 5.22 1.30 0.548 description). They do not include jitter. 2457337.7891 5.41 1.59 0.545 ( ) 2457340.95644 −1.99 1.27 0.553 This table is available in machine-readable form. 2457347.86896 4.10 1.29 0.556 2457348.77993 4.65 1.27 0.556 We ran this analysis for all of the age models. Each of the 2457350.72611 5.83 1.20 0.558 models satisfied the convergence criterion for all parameters 2457354.70613 −0.88 1.65 0.548 except for longitude of the ascending node (Ω), which is poorly 2457361.64656 17.26 1.43 0.549 constrained by our data in all cases. However, the 200 Myr
15 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al.
Figure 12. Keck/NIRC2 Ms-band vortex performance/completeness map for a τ=5σ detection threshold comparing using the SB12 evolutionary model (Spiegel & Burrows 2012). The yellow curve highlights the 95% completeness contour. The red curve highlights the 95% completeness contour for the COND model as in Figure 4. Figure 10. Same as Figure 9, but with uniform prior on companion mass.
Figure 11. Two-dimensional posterior distribution of the position of the planet during the epoch of the imaging observations. This posterior was produced using the RV likelihood only, and demonstrates that the planet was optimally Figure 13. Posterior probability distributions for the mass of ò Eridani b, when separated from its host star at the imaging epoch. The values of the pixels in the a Gaussian prior preferring orbits coplanar with the measured inclination of the maximum-likelihood annulus contribute most significantly to the direct disk (i=34°±2°) is applied instead of the uninformative geometric prior imaging likelihood. typically used. The resultant distributions are normalized in the plot. The posterior probability distributions are mutually consistent with one another model failed to converge in many other parameters, most likely because Mb sin i is well-constrained by the RV data. If coplanarity is assumed, ± due to the conflict between the youngest models and the the mass of the planet is 1.19 0.12 MJ. coplanarity condition. We therefore report only the 400 Myr and older model results here. another, with the exception of the 200 Myr model, which did The majority of the fit parameters have posterior distributions not achieve convergence. The new median and 68% credible = ± consistent with the previous analyses. Because inclination does intervals on the planet mass for each model are mb 1.19 ( ) +0.12 ( ) +0.11 not correlate strongly with any parameters except companion 0.12 MJ RV only , 1.18-0.11 MJ 800 Myr , and 1.19-0.12 MJ ( ) mass, we do not expect this new prior to affect the posterior 400 Myr . The posterior distributions are plotted in Figure 13. probability distributions for any of the other model parameters. – For all age models, the companion mass constraint changes 5.3. Planet disk Interactions significantly compared to the edge-on orbits preferred by the In this section, we investigate the possible connection uninformative prior. The new constraints are all similar to one between ò Eridani b and the system’s debris belts. Debris disks
16 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al. are the leftovers of planet formation and primarily made out of small bodies (asteroids, comets, dust) that are the remnants of planetesimals, the building blocks of planet cores. Debris disks can be traced by the intrinsic thermal emission of micron-sized particles generated in collisional cascades between the small bodies. For some favorable geometries, scattering of the small particles can also be detected at optical wavelengths. The dust is constantly replenished by the collisions between leftover planetesimals that are gravitationally stirred by themselves or by neighboring planets. Non-axisymmetric features and separated rings revealed in scattered light images and/or thermal emissions of micron-sized to millimeter-sized dust particles are considered signposts of existing planets. The connection between debris disks and planets has been seen in several of the currently known, directly imaged planetary systems, such as HR 8799, β Pictoris, HD 95086, HD 106906, Figure 14. The chaotic zone around ò Eridani b as a function of particle free ( ( ) eccentricity. The hatched zone corresponds to the chaotic zone around the most Fomalhaut, and 51 Eridani see, e.g., Bowler 2016 for a likely location of the planet (a=3.48 au; horizontal solid line). The planet recent review). Using the largest sample of debris disks systems mass is fixed to 0.78 MJup. We expect the region between ∼2.7 and ∼4.3 au to directly surveyed for long-period giant planets to date, Meshkat be cleared out by ò Eridani b. et al. (2017) recently found that the occurrence rate of long- period giant planets in dusty systems is about ten times higher than in dust-free systems (88% confidence level), providing the 5.3.2. Dust Production in Planetesimal Belts: Planet-stirred first tentative empirical evidence for a planet–disk connection. or Self-stirred? Dust grains that are millimeter-sized and smaller that 5.3.1. Constraining the Inner Belt(s) dominate the radio and scattered light images are the products of collisions among larger planetesimals. Particles can collide Armed with refined orbital parameters of ò Eridani b, we with each other as they gravitationally stir each other or they place constraints on the inner belt(s) identified by Su et al. may be secularly perturbed by a planet. Is ò Eridani b (2017). Close to the planet, inside a chaotic zone, an overlap of mean motion resonances destabilizes the orbits of particles on responsible for triggering collisional cascades in the warm and short timescales, effectively clearing them out (e.g., Wisdom cold belts detected in the system, or are the planetesimals more 1980; Quillen & Faber 2006; Morrison & Malhotra 2015). likely to be self-stirred? Wisdom (1980) showed that the width of the chaotic zone The characteristic timescale for planetesimals to stir each / ( scales with μ2 7, where μ is the mass ratio between the planet other is set by their rate of collision see, e.g., Section 4.1.4 of ( )) ( and the star. Quillen & Faber (2006) find that the width of the Goldreich et al. 2004 . Numerical e.g., Kenyon & Brom- ley 2008) and analytic (e.g., Pan & Schlichting 2012) chaotic zone is independent of planet eccentricity epl, as long as calculations suggest bodies as large as ∼1000 km are at the epl0.3. We have shown that ò Eridani b has an eccentricity consistent with zero and definitively smaller than 0.3. Using the top of the collisional cascade: they can stir up smaller bodies to numerically calibrated expressions for the chaotic zone’s inner disruption at a rate faster than their growth via accretion. It is 0.28 0.31 edge (11.17- m )apl and the outer edge (11.76+ m )apl the formation timescale of these Pluto-sized bodies that limits where apl is the semimajor axis of the planet (see Morrison & the overall self-stirring timescale: Malhotra 2015, their Table 1),wefind that ò Eridani b would clear out 2.90–4.19 au if the planet was located at 3.48 au ⎛ ⎞1.15⎛ ⎞3⎛ ⎞32 ⎜⎟⎜⎟SMMSN aMdisk ( tss ~ 173 Myr ⎜ ⎟ ,10() equivalent to a 0.78 MJup planet orbiting a star of 0.78 Me at ⎝ S ⎠ ⎝ 60 au ⎠ ⎝ M ⎠ orbital periods of 7.37 yr; see the “800 Myr” column of Table 4). fi where adisk is the semimajor axis of the planetesimals, Må is If the particles within the planetesimal belts have suf ciently -2 large free eccentricities (i.e., initial eccentricities set by self- the mass of the host star, SºMMSN 0.18 g cm ()MM stirring and collision; see, e.g., Pan & Schlichting (2012)), the (a 30 au)-32is the Minimum Mass Solar Nebula (MMSN), chaotic zone can widen, commensurate with the particle and we take the coefficient derived by numerical simulations of eccentricities (Mustill & Wyatt 2012). For ò Eridani b, the Kenyon & Bromley (2008). The calculation above assumes / critical particle eccentricity is only ∼0.21 μ3 7∼0.01. In large bodies build up mostly after the dispersal of the gas disk, Figure 14, we show that, for particles more eccentric than because gas dynamical friction tends to damp away the ∼ ò 0.01, the chaotic zone around Eridani b can widen by a few eccentricities of solids. It may be that planetesimals are built tens of percent. — ∼ rapidly even in the early stages of gas disk evolution by Overall, we expect there to be no particles between 2.7 and ( ∼ streaming instability e.g., Youdin & Goodman 2005; Johansen 4.3 au. If the excess IR emission interior to 25 au is from one ) ( ) ∼ et al. 2007 , resonant drag instability Squire & Hopkins 2018 , broad disk, its inner edge must be beyond 4.3 au. If instead / ( the excess emission is from two narrow belts, the innermost and or pebble accretion e.g., Ormel & Klahr 2010; Ormel )— belt must be inside ∼2.7 au while the outermost belt must be 2017 such that Pluto-sized bodies are already available by outside ∼4.3 au. Both scenarios are roughly consistent with the the time the disk gas dissipates. Without the need to wait for the analysis of Su et al. (2017) and the most recent LBTI results build-up of large bodies after the gas dispersal, tss can be (Ertel et al. 2018). shortened by an order of magnitude (Krivov & Booth 2018).
17 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al.
rapidly, such that our tss is overestimated. It may also be that an extra planet beyond the orbit of ò Eridani b is responsible for stirring the outermost belt and for carving out the region between the inner warm belts and the outer cold belt. Our solution with uninformative inclination priors to the orbit of ò Eridani b favors an orbital inclination (89°±42°) that is larger than the inclination of the outermost belt (34°±2°). Such a large misalignment suggests that ò Eridani may have interacted with a fly-by in the past and/or there exists another planet that is inclined to the orbit of ò Eridani b. Should this extra planet be responsible for tilting the outer belt from the orbital plane of ò Eridani b, we expect its orbit to be misaligned with respect to ò Eridani b by at least 10°, assuming the mass of the extra planet is ∼1 Jupiter mass with an orbital distance of 48 au. Figure 15. The timescale for ò Eridani b to stir particles at different distances In Meshkat et al. (2017), we found that the occurrence rate of (Equations (11) and (12)) compared against the self-stirring timescale long-period planets in debris disks is about 10 times higher (Equation (10)). The blue bar reflects the range of eccentricities of ò Eridani ò b from our MCMC analysis, with the lower and upper limit corresponding to than in dust-free systems. The Eridani system is rich in dust epl=0.13 and epl=0.02, respectively; the solid line represents the median contained in multiple belts reminiscent of the HR 8799 system epl=0.07. We fix Mpl=0.78 MJup, apl=3.48 au, and Må=0.78 Me.We (Su et al. 2009), so it may very well harbor more than one have assumed MMSN surface density to calculate the self-stirring timescale. planet, as suggested by previous studies (Mizuki et al. 2016; The expected chaotic zone of ò Eridani b is depicted with a gray zone. Three ) fi ( ) Booth et al. 2017 . Unfortunately, the eld of view of our high- orange bars illustrate the warm and cold belts reported by Su et al. 2017 and / Booth et al. (2017), respectively. Unless ò Eridani’s disk is significantly less contrast Keck NIRC2 data set is too small to probe a large massive than MMSN—by more than an order of magnitude—it is likely that separation in the vicinity of the outer disk. the collisions within the innermost warm belt and the outermost cold belt are driven by self-stirring rather than the secular perturbation by ò Eridani b. Collisions in the secondary warm belt that span 8–20 au are consistent with 5.3.3. Imaging the Elusive ò Eridani Planet(s) with JWST both self-stirring and planet-stirring. The Keck/NIRC2 Ms-band vortex coronagraph high-con- trast images presented here showcase an exquisite inner The orbit crossing timescale for two nearby planetesimals as ( ) ″ they are secularly perturbed by a planet is t ∼1/Ae where working angle 0 2 and sensitivity up to 5 . Ground-based cross p adaptive optics and small-angle coronagraphy in the mid- A is the precession frequency and ep is the eccentricity of the ( μ ) perturbed particle (Mustill & Wyatt 2009). For particles on infrared from 3 to 5 m on 10 m class telescopes will only be challenged by the advent of Giant Segmented Mirror initially circular orbits, ep is set by the forced eccentricity such ( ) that the orbit-crossing timescale becomes Telescopes GSMTs . However, NIRCam and MIRI, the infrared cameras of the upcoming James Webb Space ()1 - e 2 32⎛ ⎞92 Telescope, will have unmatched sensitivity beyond 1″ from pl ⎜⎟adisk tcross » 500 Gyr the star. Thus, JWST’s NIRCam and MIRI are ideal instruments e ⎝ 60 au ⎠ pl to explore the inner cavity of ò Eridani from 3 to 30 μm, and ⎛ ⎞12⎛ ⎞⎛ ⎞3 ″ M MJup 3.5 au reveal additional elusive planets. The outer ring being at 20 ´ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ()11 from the host star, the diffraction and scattering from the star ⎝ M ⎠ ⎝ Ma⎠⎝ ⎠ pl pl will be less of a concern, allowing us to probe the cavity for the for internal perturbers, and putative planet shepherding the outer ring. Invoking dynamical arguments similar to those used in the previous sections and ()1 - e 2 32⎛ a ⎞4 upper limits from Spitzer images (Janson et al. 2015), Booth pl ⎜⎟pl tcross » 17.4 kyr ( ) e ⎝ 3.5 au ⎠ et al. 2017 predict that the putative exoplanet responsible for pl shaping the outer belt is at a semimajor axis of 48 au, with a ⎛ ⎞12⎛ ⎞⎛ ⎞52 M MJup 1au mass between 0.4 and 1.2 MJ. ´ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ()12 ’ ⎝ M ⎠ ⎝ Ma⎠⎝ ⎠ JWST s NIRCam instrument team is planning coronagraphic pl disk observations at 4.4 μm (F444W) to search both the interior (± ″) ( ) ò for external perturbers, where epl is the planet eccentricity, Mpl is 10 and exterior regions 2 2 around Eridani for planets. the planet mass, and a is the semimajor axis of the perturbing The interior region will be observed twice to reject background pl fi planet (see Mustill & Wyatt (2009), their Equations (15) and 16). objects on the basis of source motion, while the entire eld will be observed at 3 μm to reject background stars and galaxies on For ò Eridani b to be responsible for the dust production in the basis of color. The expected contrast ratio after roll and planetesimal belts, t must be shorter than the age of the −6 cross reference star subtraction is expected to be ∼10 at 1″ and < fi ò − system and tcross tss. For our re ned parameters for Eridani ∼10 7 at 2″, depending on the in-orbit stability of JWST (Krist fi b, we nd that, out of the three belts that are imaged or inferred et al. 2007; Beichman et al. 2010). to exist, the innermost and the outermost belts are more likely As noted above, translating between instrumental detection self-stirred, while the intermediate warm belt can be either self- limits and planet mass is challenging due to the uncertainties in stirred or planet-stirred (see Figure 15). stellar age and exoplanet models at low masses. For a nominal Even the self-stirring timescale for the outer belt is age of 800 Myr, the SB12 and COND models yield F444W uncomfortably close to the system’s estimated age. It may be brightness estimates ranging from 16 to 18 mag for a 0.78 MJup that the largest bodies that trigger a collisional cascade emerge planet (effective temperature of ;150 K). In the most favorable
18 The Astronomical Journal, 157:33 (20pp), 2019 January Mawet et al. case (SB12), NIRCam could detect ò Eridani b at 1″ (close to The authors would like to acknowledge the contributions and the nominal 7.37 yr, 3.48 au orbit) with S/N ∼5 and beyond 2″ useful comments from Prof.Debra Fischer and Prof.James with S/N>25. Within 1″, the Keck observations reported Graham. The data presented herein were obtained at the W.M. here are comparable to or more sensitive than planned JWST Keck Observatory, which is operated as a scientific partnership observations due to Keck’s larger aperture and the improved among the California Institute of Technology, the University of performance of the vortex coronagraph. At larger separations, California, and the National Aeronautics and Space Adminis- however, the low thermal background in space gives JWST a tration (NASA). The Observatory was made possible by the fi significant advantage to look for <0.5 MJup planets (depending generous nancial support of the W.M. Keck Foundation. The on model assumptions) that might be responsible for the disk authors wish to recognize and acknowledge the very significant structures seen in the far-infrared and by ALMA. Because the cultural role and reverence that the summit of Maunakea has field of view of both NIRCam and MIRI is 20″×20″, at least always had within the indigenous Hawaiian community. We two pointings will be necessary to map out the inner cavity at are most fortunate to have the opportunity to conduct the vicinity of the outer ring. observations from this mountain. This work was partially performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. (C) 2017. All rights 6. Conclusion reserved. E.C. acknowledges support from NASA through Hubble Fellowship grant HF2-51355 awarded by STScI, which In this paper, we have presented the most sensitive and is operated by AURA, Inc. for NASA under contract NAS5- comprehensive observational evidence for the existence of 26555. We also acknowledge support from NASA/NExSS ò Eridani b. Combining exquisite RV and direct imaging through grant no. NNX15AD95G. data provides unprecedented constraints on the mass and Software:RadVel (Fulton et al. 2018), emcee (Foreman- orbital parameters of the planet. ò Eridani b has a mass of Mackey et al. 2013), VIP (Gomez Gonzalez et al. 2017), +0.38 ò ± PyKLIP ( ) 0.78-0.12 MJup and is orbiting Eridani at about 3.48 0.02 au Wang et al. 2015 . in 7.37±0.07 yr. Assuming coplanarity with the outer belt resolved by ALMA (inclination of 34°±2°), the mass of the ORCID iDs planet is approximately 1.19 M . Our data and analysis also Jup // / present compelling lines of evidence that the system’s age is Dimitri Mawet https: orcid.org 0000-0002-8895-4735 // / closer to 800 Myr or more. Indeed, the direct imaging data Lea Hirsch https: orcid.org 0000-0001-8058-7443 ’ Eve J. Lee https://orcid.org/0000-0002-1228-9820 should have shown the planet if the system s age was 200 Myr. fi // / Notably, we also find that the eccentricity of ò Eridani b is a Jean-Baptiste Ruf o https: orcid.org 0000-0003- very low, 0.07+0.06, an order of magnitude smaller than early 2233-4821 -0.05 Michael Bottom https://orcid.org/0000-0003-1341-5531 estimates and consistent with a circular orbit. For that reason Benjamin J. Fulton https://orcid.org/0000-0003-3504-5316 and using simple dynamical arguments, we postulate that Olivier Absil https://orcid.org/0000-0002-4006-6237 ò Eridani b is unlikely to be responsible for stirring the outer Brendan Bowler https://orcid.org/0000-0003-2649-2288 debris belt at 70 au, and that one or more additional planets Marta Bryan https://orcid.org/0000-0002-6076-5967 must be shepherding it. However, ò Eridani b could be // / ( ) Elodie Choquet https: orcid.org 0000-0002-9173-0740 responsible for shaping the putative warm belt s within 25 au, Denis Defrère https://orcid.org/0000-0003-3499-2506 although self-stirring is another likely dust production mech- Carlos Alberto Gomez Gonzalez https://orcid.org/0000- anism. If any additional planets exist, they will be easily 0003-2050-1710 ’ μ detected with JWST s NIRCam and MIRI from 4 to 25 m, Andrew W. Howard https://orcid.org/0000-0001- enabling unique spectroscopic and dynamical characterization 8638-0320 opportunities. Howard Isaacson https://orcid.org/0000-0002-0531-1073 This paper also demonstrates the unique power of the Rebecca Jensen-Clem https://orcid.org/0000-0003- combination of RV and direct imaging observations to detect 0054-2953 and constrain the masses of giant planets within 10 au. The Molly Kosiarek https://orcid.org/0000-0002-6115-4359 long history of RV observations limited the spatial domain in Geoff Marcy https://orcid.org/0000-0002-2909-0113 which ò Eridani b could reside, allowing optimized detect- Tiffany Meshkat https://orcid.org/0000-0001-6126-2467 ability with the imaging observations. Erik Petigura https://orcid.org/0000-0003-0967-2893 We would be remiss if we did not mention as a caveat that Maddalena Reggiani https://orcid.org/0000-0003- the direct imaging nondetection of the planet leaves open the 2911-0898 possibility, however small, that the planet does not actually Garreth Ruane https://orcid.org/0000-0003-4769-1665 exist. Although there is no obvious commensurability between Evan Sinukoff https://orcid.org/0000-0002-5658-0601 the long-term magnetic activity diagnostic (SHK) and the RV Ji Wang https://orcid.org/0000-0002-4361-8885 data, as demonstrated in Section 4.2, it remains possible that Lauren Weiss https://orcid.org/0000-0002-3725-3058 the magnetic field affects the RVs in a way we do not understand. References ò Eridani remains a fascinating testbed for studying planetary formation in great detail. We conclude this paper by Anglada-Escudé, G., & Butler, R. P. 2012, ApJS, 200, 15 emphasizing the perfect complementarity between long-term Aumann, H. H. 1985, PASP, 97, 885 RV monitoring, mid-infrared small-angle high-contrast ground- Backman, D., Marengo, M., Stapelfeldt, K., et al. 2009, ApJ, 690, 1522 Baraffe, I., Chabrier, G., Barman, T. S., Allard, F., & Hauschildt, P. H. 2003, based capabilities, and the sensitive space-based parameter A&A, 402, 701 space to be opened by JWST. Beichman, C. A., Krist, J., Trauger, J. T., et al. 2010, PASP, 122, 162
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